Consider the metric for an arbitrary spherically symmetric space-time (of, say, a collapsing or exploding star,...

 

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Consider the metric for an arbitrary spherically symmetric space-time (of, say, a collapsing or exploding star, or a black hole): ds² = 9da"da" = -e²0dT²+e²dR² + r²(d0² + sin² 0 do²) where $=$(T, R) A = A(T, R) r=r(T, R) are as-yet-unspecified functions of T and R only. (a) COMPUTE (using the Cartan-Misner approach) i. the connection 1-forms relative to the orthonormal basis ii. the non-zero orthonormal basis components of the Riemann curvature tensor. Helpful suggestion # 1: Denote partial derivatives by dots and primes, e.g. әф მ '; etc. ar ƏR Helpful suggestion # 2: Make sure your result(s) is (are) exhibited prominently. (b) FIND the coordinate basis components of Riemann in terms of the o.n. components found in (a). Consider the metric for an arbitrary spherically symmetric space-time (of, say, a collapsing or exploding star, or a black hole): ds² = 9da"da" = -e²0dT²+e²dR² + r²(d0² + sin² 0 do²) where $=$(T, R) A = A(T, R) r=r(T, R) are as-yet-unspecified functions of T and R only. (a) COMPUTE (using the Cartan-Misner approach) i. the connection 1-forms relative to the orthonormal basis ii. the non-zero orthonormal basis components of the Riemann curvature tensor. Helpful suggestion # 1: Denote partial derivatives by dots and primes, e.g. әф მ '; etc. ar ƏR Helpful suggestion # 2: Make sure your result(s) is (are) exhibited prominently. (b) FIND the coordinate basis components of Riemann in terms of the o.n. components found in (a).

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Answer a i Connection 1forms We first compute the components of the metric g00 e2 g11 e2A g22 r2 g33 ... View the full answer